/* * Copyright 2011 Google Inc. * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ #include "GrPathUtils.h" #include "GrTypes.h" #include "SkGeometry.h" #include "SkMathPriv.h" static const int MAX_POINTS_PER_CURVE = 1 << 10; static const SkScalar gMinCurveTol = 0.0001f; SkScalar GrPathUtils::scaleToleranceToSrc(SkScalar devTol, const SkMatrix& viewM, const SkRect& pathBounds) { // In order to tesselate the path we get a bound on how much the matrix can // scale when mapping to screen coordinates. SkScalar stretch = viewM.getMaxScale(); if (stretch < 0) { // take worst case mapRadius amoung four corners. // (less than perfect) for (int i = 0; i < 4; ++i) { SkMatrix mat; mat.setTranslate((i % 2) ? pathBounds.fLeft : pathBounds.fRight, (i < 2) ? pathBounds.fTop : pathBounds.fBottom); mat.postConcat(viewM); stretch = SkMaxScalar(stretch, mat.mapRadius(SK_Scalar1)); } } SkScalar srcTol = devTol / stretch; if (srcTol < gMinCurveTol) { srcTol = gMinCurveTol; } return srcTol; } uint32_t GrPathUtils::quadraticPointCount(const SkPoint points[], SkScalar tol) { // You should have called scaleToleranceToSrc, which guarantees this SkASSERT(tol >= gMinCurveTol); SkScalar d = points[1].distanceToLineSegmentBetween(points[0], points[2]); if (!SkScalarIsFinite(d)) { return MAX_POINTS_PER_CURVE; } else if (d <= tol) { return 1; } else { // Each time we subdivide, d should be cut in 4. So we need to // subdivide x = log4(d/tol) times. x subdivisions creates 2^(x) // points. // 2^(log4(x)) = sqrt(x); SkScalar divSqrt = SkScalarSqrt(d / tol); if (((SkScalar)SK_MaxS32) <= divSqrt) { return MAX_POINTS_PER_CURVE; } else { int temp = SkScalarCeilToInt(divSqrt); int pow2 = GrNextPow2(temp); // Because of NaNs & INFs we can wind up with a degenerate temp // such that pow2 comes out negative. Also, our point generator // will always output at least one pt. if (pow2 < 1) { pow2 = 1; } return SkTMin(pow2, MAX_POINTS_PER_CURVE); } } } uint32_t GrPathUtils::generateQuadraticPoints(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, SkScalar tolSqd, SkPoint** points, uint32_t pointsLeft) { if (pointsLeft < 2 || (p1.distanceToLineSegmentBetweenSqd(p0, p2)) < tolSqd) { (*points)[0] = p2; *points += 1; return 1; } SkPoint q[] = { { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, }; SkPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }; pointsLeft >>= 1; uint32_t a = generateQuadraticPoints(p0, q[0], r, tolSqd, points, pointsLeft); uint32_t b = generateQuadraticPoints(r, q[1], p2, tolSqd, points, pointsLeft); return a + b; } uint32_t GrPathUtils::cubicPointCount(const SkPoint points[], SkScalar tol) { // You should have called scaleToleranceToSrc, which guarantees this SkASSERT(tol >= gMinCurveTol); SkScalar d = SkTMax( points[1].distanceToLineSegmentBetweenSqd(points[0], points[3]), points[2].distanceToLineSegmentBetweenSqd(points[0], points[3])); d = SkScalarSqrt(d); if (!SkScalarIsFinite(d)) { return MAX_POINTS_PER_CURVE; } else if (d <= tol) { return 1; } else { SkScalar divSqrt = SkScalarSqrt(d / tol); if (((SkScalar)SK_MaxS32) <= divSqrt) { return MAX_POINTS_PER_CURVE; } else { int temp = SkScalarCeilToInt(SkScalarSqrt(d / tol)); int pow2 = GrNextPow2(temp); // Because of NaNs & INFs we can wind up with a degenerate temp // such that pow2 comes out negative. Also, our point generator // will always output at least one pt. if (pow2 < 1) { pow2 = 1; } return SkTMin(pow2, MAX_POINTS_PER_CURVE); } } } uint32_t GrPathUtils::generateCubicPoints(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, const SkPoint& p3, SkScalar tolSqd, SkPoint** points, uint32_t pointsLeft) { if (pointsLeft < 2 || (p1.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd && p2.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd)) { (*points)[0] = p3; *points += 1; return 1; } SkPoint q[] = { { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) } }; SkPoint r[] = { { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }, { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) } }; SkPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) }; pointsLeft >>= 1; uint32_t a = generateCubicPoints(p0, q[0], r[0], s, tolSqd, points, pointsLeft); uint32_t b = generateCubicPoints(s, r[1], q[2], p3, tolSqd, points, pointsLeft); return a + b; } int GrPathUtils::worstCasePointCount(const SkPath& path, int* subpaths, SkScalar tol) { // You should have called scaleToleranceToSrc, which guarantees this SkASSERT(tol >= gMinCurveTol); int pointCount = 0; *subpaths = 1; bool first = true; SkPath::Iter iter(path, false); SkPath::Verb verb; SkPoint pts[4]; while ((verb = iter.next(pts)) != SkPath::kDone_Verb) { switch (verb) { case SkPath::kLine_Verb: pointCount += 1; break; case SkPath::kConic_Verb: { SkScalar weight = iter.conicWeight(); SkAutoConicToQuads converter; const SkPoint* quadPts = converter.computeQuads(pts, weight, tol); for (int i = 0; i < converter.countQuads(); ++i) { pointCount += quadraticPointCount(quadPts + 2*i, tol); } } case SkPath::kQuad_Verb: pointCount += quadraticPointCount(pts, tol); break; case SkPath::kCubic_Verb: pointCount += cubicPointCount(pts, tol); break; case SkPath::kMove_Verb: pointCount += 1; if (!first) { ++(*subpaths); } break; default: break; } first = false; } return pointCount; } void GrPathUtils::QuadUVMatrix::set(const SkPoint qPts[3]) { SkMatrix m; // We want M such that M * xy_pt = uv_pt // We know M * control_pts = [0 1/2 1] // [0 0 1] // [1 1 1] // And control_pts = [x0 x1 x2] // [y0 y1 y2] // [1 1 1 ] // We invert the control pt matrix and post concat to both sides to get M. // Using the known form of the control point matrix and the result, we can // optimize and improve precision. double x0 = qPts[0].fX; double y0 = qPts[0].fY; double x1 = qPts[1].fX; double y1 = qPts[1].fY; double x2 = qPts[2].fX; double y2 = qPts[2].fY; double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2; if (!sk_float_isfinite(det) || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) { // The quad is degenerate. Hopefully this is rare. Find the pts that are // farthest apart to compute a line (unless it is really a pt). SkScalar maxD = qPts[0].distanceToSqd(qPts[1]); int maxEdge = 0; SkScalar d = qPts[1].distanceToSqd(qPts[2]); if (d > maxD) { maxD = d; maxEdge = 1; } d = qPts[2].distanceToSqd(qPts[0]); if (d > maxD) { maxD = d; maxEdge = 2; } // We could have a tolerance here, not sure if it would improve anything if (maxD > 0) { // Set the matrix to give (u = 0, v = distance_to_line) SkVector lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge]; // when looking from the point 0 down the line we want positive // distances to be to the left. This matches the non-degenerate // case. lineVec.setOrthog(lineVec, SkPoint::kLeft_Side); // first row fM[0] = 0; fM[1] = 0; fM[2] = 0; // second row fM[3] = lineVec.fX; fM[4] = lineVec.fY; fM[5] = -lineVec.dot(qPts[maxEdge]); } else { // It's a point. It should cover zero area. Just set the matrix such // that (u, v) will always be far away from the quad. fM[0] = 0; fM[1] = 0; fM[2] = 100.f; fM[3] = 0; fM[4] = 0; fM[5] = 100.f; } } else { double scale = 1.0/det; // compute adjugate matrix double a2, a3, a4, a5, a6, a7, a8; a2 = x1*y2-x2*y1; a3 = y2-y0; a4 = x0-x2; a5 = x2*y0-x0*y2; a6 = y0-y1; a7 = x1-x0; a8 = x0*y1-x1*y0; // this performs the uv_pts*adjugate(control_pts) multiply, // then does the scale by 1/det afterwards to improve precision m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale); m[SkMatrix::kMSkewX] = (float)((0.5*a4 + a7)*scale); m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale); m[SkMatrix::kMSkewY] = (float)(a6*scale); m[SkMatrix::kMScaleY] = (float)(a7*scale); m[SkMatrix::kMTransY] = (float)(a8*scale); // kMPersp0 & kMPersp1 should algebraically be zero m[SkMatrix::kMPersp0] = 0.0f; m[SkMatrix::kMPersp1] = 0.0f; m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale); // It may not be normalized to have 1.0 in the bottom right float m33 = m.get(SkMatrix::kMPersp2); if (1.f != m33) { m33 = 1.f / m33; fM[0] = m33 * m.get(SkMatrix::kMScaleX); fM[1] = m33 * m.get(SkMatrix::kMSkewX); fM[2] = m33 * m.get(SkMatrix::kMTransX); fM[3] = m33 * m.get(SkMatrix::kMSkewY); fM[4] = m33 * m.get(SkMatrix::kMScaleY); fM[5] = m33 * m.get(SkMatrix::kMTransY); } else { fM[0] = m.get(SkMatrix::kMScaleX); fM[1] = m.get(SkMatrix::kMSkewX); fM[2] = m.get(SkMatrix::kMTransX); fM[3] = m.get(SkMatrix::kMSkewY); fM[4] = m.get(SkMatrix::kMScaleY); fM[5] = m.get(SkMatrix::kMTransY); } } } //////////////////////////////////////////////////////////////////////////////// // k = (y2 - y0, x0 - x2, x2*y0 - x0*y2) // l = (y1 - y0, x0 - x1, x1*y0 - x0*y1) * 2*w // m = (y2 - y1, x1 - x2, x2*y1 - x1*y2) * 2*w void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkMatrix* out) { SkMatrix& klm = *out; const SkScalar w2 = 2.f * weight; klm[0] = p[2].fY - p[0].fY; klm[1] = p[0].fX - p[2].fX; klm[2] = p[2].fX * p[0].fY - p[0].fX * p[2].fY; klm[3] = w2 * (p[1].fY - p[0].fY); klm[4] = w2 * (p[0].fX - p[1].fX); klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY); klm[6] = w2 * (p[2].fY - p[1].fY); klm[7] = w2 * (p[1].fX - p[2].fX); klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY); // scale the max absolute value of coeffs to 10 SkScalar scale = 0.f; for (int i = 0; i < 9; ++i) { scale = SkMaxScalar(scale, SkScalarAbs(klm[i])); } SkASSERT(scale > 0.f); scale = 10.f / scale; for (int i = 0; i < 9; ++i) { klm[i] *= scale; } } //////////////////////////////////////////////////////////////////////////////// namespace { // a is the first control point of the cubic. // ab is the vector from a to the second control point. // dc is the vector from the fourth to the third control point. // d is the fourth control point. // p is the candidate quadratic control point. // this assumes that the cubic doesn't inflect and is simple bool is_point_within_cubic_tangents(const SkPoint& a, const SkVector& ab, const SkVector& dc, const SkPoint& d, SkPathPriv::FirstDirection dir, const SkPoint p) { SkVector ap = p - a; SkScalar apXab = ap.cross(ab); if (SkPathPriv::kCW_FirstDirection == dir) { if (apXab > 0) { return false; } } else { SkASSERT(SkPathPriv::kCCW_FirstDirection == dir); if (apXab < 0) { return false; } } SkVector dp = p - d; SkScalar dpXdc = dp.cross(dc); if (SkPathPriv::kCW_FirstDirection == dir) { if (dpXdc < 0) { return false; } } else { SkASSERT(SkPathPriv::kCCW_FirstDirection == dir); if (dpXdc > 0) { return false; } } return true; } void convert_noninflect_cubic_to_quads(const SkPoint p[4], SkScalar toleranceSqd, bool constrainWithinTangents, SkPathPriv::FirstDirection dir, SkTArray* quads, int sublevel = 0) { // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is // p[2]. Point d is always p[3]. Point c is p[2] unless p[2] == p[3], in which case it is p[1]. SkVector ab = p[1] - p[0]; SkVector dc = p[2] - p[3]; if (ab.lengthSqd() < SK_ScalarNearlyZero) { if (dc.lengthSqd() < SK_ScalarNearlyZero) { SkPoint* degQuad = quads->push_back_n(3); degQuad[0] = p[0]; degQuad[1] = p[0]; degQuad[2] = p[3]; return; } ab = p[2] - p[0]; } if (dc.lengthSqd() < SK_ScalarNearlyZero) { dc = p[1] - p[3]; } // When the ab and cd tangents are degenerate or nearly parallel with vector from d to a the // constraint that the quad point falls between the tangents becomes hard to enforce and we are // likely to hit the max subdivision count. However, in this case the cubic is approaching a // line and the accuracy of the quad point isn't so important. We check if the two middle cubic // control points are very close to the baseline vector. If so then we just pick quadratic // points on the control polygon. if (constrainWithinTangents) { SkVector da = p[0] - p[3]; bool doQuads = dc.lengthSqd() < SK_ScalarNearlyZero || ab.lengthSqd() < SK_ScalarNearlyZero; if (!doQuads) { SkScalar invDALengthSqd = da.lengthSqd(); if (invDALengthSqd > SK_ScalarNearlyZero) { invDALengthSqd = SkScalarInvert(invDALengthSqd); // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a. // same goes for point c using vector cd. SkScalar detABSqd = ab.cross(da); detABSqd = SkScalarSquare(detABSqd); SkScalar detDCSqd = dc.cross(da); detDCSqd = SkScalarSquare(detDCSqd); if (detABSqd * invDALengthSqd < toleranceSqd && detDCSqd * invDALengthSqd < toleranceSqd) { doQuads = true; } } } if (doQuads) { SkPoint b = p[0] + ab; SkPoint c = p[3] + dc; SkPoint mid = b + c; mid.scale(SK_ScalarHalf); // Insert two quadratics to cover the case when ab points away from d and/or dc // points away from a. if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) { SkPoint* qpts = quads->push_back_n(6); qpts[0] = p[0]; qpts[1] = b; qpts[2] = mid; qpts[3] = mid; qpts[4] = c; qpts[5] = p[3]; } else { SkPoint* qpts = quads->push_back_n(3); qpts[0] = p[0]; qpts[1] = mid; qpts[2] = p[3]; } return; } } static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; static const int kMaxSubdivs = 10; ab.scale(kLengthScale); dc.scale(kLengthScale); // e0 and e1 are extrapolations along vectors ab and dc. SkVector c0 = p[0]; c0 += ab; SkVector c1 = p[3]; c1 += dc; SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1); if (dSqd < toleranceSqd) { SkPoint cAvg = c0; cAvg += c1; cAvg.scale(SK_ScalarHalf); bool subdivide = false; if (constrainWithinTangents && !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) { // choose a new cAvg that is the intersection of the two tangent lines. ab.setOrthog(ab); SkScalar z0 = -ab.dot(p[0]); dc.setOrthog(dc); SkScalar z1 = -dc.dot(p[3]); cAvg.fX = ab.fY * z1 - z0 * dc.fY; cAvg.fY = z0 * dc.fX - ab.fX * z1; SkScalar z = ab.fX * dc.fY - ab.fY * dc.fX; z = SkScalarInvert(z); cAvg.fX *= z; cAvg.fY *= z; if (sublevel <= kMaxSubdivs) { SkScalar d0Sqd = c0.distanceToSqd(cAvg); SkScalar d1Sqd = c1.distanceToSqd(cAvg); // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know // the distances and tolerance can't be negative. // (d0 + d1)^2 > toleranceSqd // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd SkScalar d0d1 = SkScalarSqrt(d0Sqd * d1Sqd); subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd; } } if (!subdivide) { SkPoint* pts = quads->push_back_n(3); pts[0] = p[0]; pts[1] = cAvg; pts[2] = p[3]; return; } } SkPoint choppedPts[7]; SkChopCubicAtHalf(p, choppedPts); convert_noninflect_cubic_to_quads(choppedPts + 0, toleranceSqd, constrainWithinTangents, dir, quads, sublevel + 1); convert_noninflect_cubic_to_quads(choppedPts + 3, toleranceSqd, constrainWithinTangents, dir, quads, sublevel + 1); } } void GrPathUtils::convertCubicToQuads(const SkPoint p[4], SkScalar tolScale, SkTArray* quads) { SkPoint chopped[10]; int count = SkChopCubicAtInflections(p, chopped); const SkScalar tolSqd = SkScalarSquare(tolScale); for (int i = 0; i < count; ++i) { SkPoint* cubic = chopped + 3*i; // The direction param is ignored if the third param is false. convert_noninflect_cubic_to_quads(cubic, tolSqd, false, SkPathPriv::kCCW_FirstDirection, quads); } } void GrPathUtils::convertCubicToQuadsConstrainToTangents(const SkPoint p[4], SkScalar tolScale, SkPathPriv::FirstDirection dir, SkTArray* quads) { SkPoint chopped[10]; int count = SkChopCubicAtInflections(p, chopped); const SkScalar tolSqd = SkScalarSquare(tolScale); for (int i = 0; i < count; ++i) { SkPoint* cubic = chopped + 3*i; convert_noninflect_cubic_to_quads(cubic, tolSqd, true, dir, quads); } } //////////////////////////////////////////////////////////////////////////////// /** * Computes an SkMatrix that can find the cubic KLM functionals as follows: * * | ..K.. | | ..kcoeffs.. | * | ..L.. | = | ..lcoeffs.. | * inverse_transpose_power_basis_matrix * | ..M.. | | ..mcoeffs.. | * * 'kcoeffs' are the power basis coefficients to a scalar valued cubic function that returns the * signed distance to line K from a given point on the curve: * * k(t,s) = C(t,s) * K [C(t,s) is defined in the following comment] * * The same applies for lcoeffs and mcoeffs. These are found separately, depending on the type of * curve. There are 4 coefficients but 3 rows in the matrix, so in order to do this calculation the * caller must first remove a specific column of coefficients. * * @return which column of klm coefficients to exclude from the calculation. */ static int calc_inverse_transpose_power_basis_matrix(const SkPoint pts[4], SkMatrix* out) { using SkScalar4 = SkNx<4, SkScalar>; // First we convert the bezier coordinates 'pts' to power basis coefficients X,Y,W=[0 0 0 1]. // M3 is the matrix that does this conversion. The homogeneous equation for the cubic becomes: // // | X Y 0 | // C(t,s) = [t^3 t^2*s t*s^2 s^3] * | . . 0 | // | . . 0 | // | . . 1 | // const SkScalar4 M3[3] = {SkScalar4(-1, 3, -3, 1), SkScalar4(3, -6, 3, 0), SkScalar4(-3, 3, 0, 0)}; // 4th column of M3 = SkScalar4(1, 0, 0, 0)}; SkScalar4 X(pts[3].x(), 0, 0, 0); SkScalar4 Y(pts[3].y(), 0, 0, 0); for (int i = 2; i >= 0; --i) { X += M3[i] * pts[i].x(); Y += M3[i] * pts[i].y(); } // The matrix is 3x4. In order to invert it, we first need to make it square by throwing out one // of the top three rows. We toss the row that leaves us with the largest absolute determinant. // Since the right column will be [0 0 1], the determinant reduces to x0*y1 - y0*x1. SkScalar absDet[4]; const SkScalar4 DETX1 = SkNx_shuffle<1,0,0,3>(X), DETY1 = SkNx_shuffle<1,0,0,3>(Y); const SkScalar4 DETX2 = SkNx_shuffle<2,2,1,3>(X), DETY2 = SkNx_shuffle<2,2,1,3>(Y); const SkScalar4 DET = DETX1 * DETY2 - DETY1 * DETX2; DET.abs().store(absDet); const int skipRow = absDet[0] > absDet[2] ? (absDet[0] > absDet[1] ? 0 : 1) : (absDet[1] > absDet[2] ? 1 : 2); const SkScalar rdet = 1 / DET[skipRow]; const int row0 = (0 != skipRow) ? 0 : 1; const int row1 = (2 == skipRow) ? 1 : 2; // Compute the inverse-transpose of the power basis matrix with the 'skipRow'th row removed. // Since W=[0 0 0 1], it follows that our corresponding solution will be equal to: // // | y1 -x1 x1*y2 - y1*x2 | // 1/det * | -y0 x0 -x0*y2 + y0*x2 | // | 0 0 det | // const SkScalar4 R(rdet, rdet, rdet, 1); X *= R; Y *= R; SkScalar x[4], y[4], z[4]; X.store(x); Y.store(y); (X * SkNx_shuffle<3,3,3,3>(Y) - Y * SkNx_shuffle<3,3,3,3>(X)).store(z); out->setAll( y[row1], -x[row1], z[row1], -y[row0], x[row0], -z[row0], 0, 0, 1); return skipRow; } static void negate_kl(SkMatrix* klm) { // We could use klm->postScale(-1, -1), but it ends up doing a full matrix multiply. for (int i = 0; i < 6; ++i) { (*klm)[i] = -(*klm)[i]; } } static void calc_serp_klm(const SkPoint pts[4], const SkScalar d[4], SkMatrix* klm) { SkMatrix CIT; int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); SkASSERT(d[0] >= 0); const SkScalar root = SkScalarSqrt(3 * d[0]); const SkScalar tl = 3 * d[2] + root; const SkScalar sl = 6 * d[1]; const SkScalar tm = 3 * d[2] - root; const SkScalar sm = 6 * d[1]; SkMatrix klmCoeffs; int col = 0; if (0 != skipCol) { klmCoeffs[0] = 0; klmCoeffs[3] = -sl * sl * sl; klmCoeffs[6] = -sm * sm * sm; ++col; } if (1 != skipCol) { klmCoeffs[col + 0] = sl * sm; klmCoeffs[col + 3] = 3 * sl * sl * tl; klmCoeffs[col + 6] = 3 * sm * sm * tm; ++col; } if (2 != skipCol) { klmCoeffs[col + 0] = -tl * sm - tm * sl; klmCoeffs[col + 3] = -3 * sl * tl * tl; klmCoeffs[col + 6] = -3 * sm * tm * tm; ++col; } SkASSERT(2 == col); klmCoeffs[2] = tl * tm; klmCoeffs[5] = tl * tl * tl; klmCoeffs[8] = tm * tm * tm; klm->setConcat(klmCoeffs, CIT); // If d1 > 0 we need to flip the orientation of our curve // This is done by negating the k and l values // We want negative distance values to be on the inside if (d[1] > 0) { negate_kl(klm); } } static void calc_loop_klm(const SkPoint pts[4], SkScalar d1, SkScalar td, SkScalar sd, SkScalar te, SkScalar se, SkMatrix* klm) { SkMatrix CIT; int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); const SkScalar tesd = te * sd; const SkScalar tdse = td * se; SkMatrix klmCoeffs; int col = 0; if (0 != skipCol) { klmCoeffs[0] = 0; klmCoeffs[3] = -sd * sd * se; klmCoeffs[6] = -se * se * sd; ++col; } if (1 != skipCol) { klmCoeffs[col + 0] = sd * se; klmCoeffs[col + 3] = sd * (2 * tdse + tesd); klmCoeffs[col + 6] = se * (2 * tesd + tdse); ++col; } if (2 != skipCol) { klmCoeffs[col + 0] = -tdse - tesd; klmCoeffs[col + 3] = -td * (tdse + 2 * tesd); klmCoeffs[col + 6] = -te * (tesd + 2 * tdse); ++col; } SkASSERT(2 == col); klmCoeffs[2] = td * te; klmCoeffs[5] = td * td * te; klmCoeffs[8] = te * te * td; klm->setConcat(klmCoeffs, CIT); // For the general loop curve, we flip the orientation in the same pattern as the serp case // above. Thus we only check d1. Technically we should check the value of the hessian as well // cause we care about the sign of d1*Hessian. However, the Hessian is always negative outside // the loop section and positive inside. We take care of the flipping for the loop sections // later on. if (d1 > 0) { negate_kl(klm); } } // For the case when we have a cusp at a parameter value of infinity (discr == 0, d1 == 0). static void calc_inf_cusp_klm(const SkPoint pts[4], SkScalar d2, SkScalar d3, SkMatrix* klm) { SkMatrix CIT; int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); const SkScalar tn = d3; const SkScalar sn = 3 * d2; SkMatrix klmCoeffs; int col = 0; if (0 != skipCol) { klmCoeffs[0] = 0; klmCoeffs[3] = -sn * sn * sn; ++col; } if (1 != skipCol) { klmCoeffs[col + 0] = 0; klmCoeffs[col + 3] = 3 * sn * sn * tn; ++col; } if (2 != skipCol) { klmCoeffs[col + 0] = -sn; klmCoeffs[col + 3] = -3 * sn * tn * tn; ++col; } SkASSERT(2 == col); klmCoeffs[2] = tn; klmCoeffs[5] = tn * tn * tn; klmCoeffs[6] = 0; klmCoeffs[7] = 0; klmCoeffs[8] = 1; klm->setConcat(klmCoeffs, CIT); } // For the case when a cubic bezier is actually a quadratic. We duplicate k in l so that the // implicit becomes: // // k^3 - l*m == k^3 - l*k == k * (k^2 - l) // // In the quadratic case we can simply assign fixed values at each control point: // // | ..K.. | | pts[0] pts[1] pts[2] pts[3] | | 0 1/3 2/3 1 | // | ..L.. | * | . . . . | == | 0 0 1/3 1 | // | ..K.. | | 1 1 1 1 | | 0 1/3 2/3 1 | // static void calc_quadratic_klm(const SkPoint pts[4], SkScalar d3, SkMatrix* klm) { SkMatrix klmAtPts; klmAtPts.setAll(0, 1.f/3, 1, 0, 0, 1, 0, 1.f/3, 1); SkMatrix inversePts; inversePts.setAll(pts[0].x(), pts[1].x(), pts[3].x(), pts[0].y(), pts[1].y(), pts[3].y(), 1, 1, 1); SkAssertResult(inversePts.invert(&inversePts)); klm->setConcat(klmAtPts, inversePts); // If d3 > 0 we need to flip the orientation of our curve // This is done by negating the k and l values if (d3 > 0) { negate_kl(klm); } } // For the case when a cubic bezier is actually a line. We set K=0, L=1, M=-line, which results in // the following implicit: // // k^3 - l*m == 0^3 - 1*(-line) == -(-line) == line // static void calc_line_klm(const SkPoint pts[4], SkMatrix* klm) { SkScalar ny = pts[0].x() - pts[3].x(); SkScalar nx = pts[3].y() - pts[0].y(); SkScalar k = nx * pts[0].x() + ny * pts[0].y(); klm->setAll( 0, 0, 0, 0, 0, 1, -nx, -ny, k); } int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkMatrix* klm, int* loopIndex) { // Variables to store the two parametric values at the loop double point. SkScalar t1 = 0, t2 = 0; // Homogeneous parametric values at the loop double point. SkScalar td, sd, te, se; SkScalar d[4]; SkCubicType cType = SkClassifyCubic(src, d); int chop_count = 0; if (SkCubicType::kLoop == cType) { SkASSERT(d[0] < 0); const SkScalar tempSqrt = SkScalarSqrt(-d[0]); td = d[2] + tempSqrt; sd = 2.f * d[1]; te = d[2] - tempSqrt; se = 2.f * d[1]; t1 = td / sd; t2 = te / se; // need to have t values sorted since this is what is expected by SkChopCubicAt if (t1 > t2) { SkTSwap(t1, t2); } SkScalar chop_ts[2]; if (t1 > 0.f && t1 < 1.f) { chop_ts[chop_count++] = t1; } if (t2 > 0.f && t2 < 1.f) { chop_ts[chop_count++] = t2; } if(dst) { SkChopCubicAt(src, dst, chop_ts, chop_count); } } else { if (dst) { memcpy(dst, src, sizeof(SkPoint) * 4); } } if (loopIndex) { if (2 == chop_count) { *loopIndex = 1; } else if (1 == chop_count) { if (t1 < 0.f) { *loopIndex = 0; } else { *loopIndex = 1; } } else { if (t1 < 0.f && t2 > 1.f) { *loopIndex = 0; } else { *loopIndex = -1; } } } if (klm) { switch (cType) { case SkCubicType::kSerpentine: case SkCubicType::kLocalCusp: calc_serp_klm(src, d, klm); break; case SkCubicType::kLoop: calc_loop_klm(src, d[1], td, sd, te, se, klm); break; case SkCubicType::kInfiniteCusp: calc_inf_cusp_klm(src, d[2], d[3], klm); break; case SkCubicType::kQuadratic: calc_quadratic_klm(src, d[3], klm); break; case SkCubicType::kLineOrPoint: calc_line_klm(src, klm); break; }; } return chop_count + 1; }